-
Notifications
You must be signed in to change notification settings - Fork 41
/
Copy pathrbf_interp_nd.html
288 lines (251 loc) · 7.98 KB
/
rbf_interp_nd.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
<html>
<head>
<title>
RBF_INTERP_ND - Multidimensional Interpolation with Radial Basis Functions
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
RBF_INTERP_ND <br> Multidimensional Interpolation with Radial Basis Functions
</h1>
<hr>
<p>
<b>RBF_INTERP_ND</b>
is a C++ library which
defines and evaluates radial basis function (RBF) interpolants to multidimensional data.
</p>
<p>
A radial basis interpolant is a useful, but expensive, technique for
definining a smooth function which interpolates a set of function values
specified at an arbitrary set of data points.
</p>
<p>
Given nd multidimensional points xd with function values fd, and a
basis function phi(r), the form of the interpolant is
<pre>
f(x) = sum ( 1 <= i <= nd ) w(i) * phi(||x-xd(i)||)
</pre>
where the weights w have been precomputed by solving
<pre>
sum ( 1 <= i <= nd ) w(i) * phi(||xd(j)-xd(i)||) = fd(j)
</pre>
</p>
<p>
Four families of radial basis functions are provided.
<ul>
<li>
phi1(r) = sqrt ( r^2 + r0^2 ) (multiquadric)
</li>
<li>
phi2(r) = 1 / sqrt ( r^2 + r0^2 ) (inverse multiquadric)
</li>
<li>
phi3(r) = r^2 * log ( r / r0 ) (thin plate spline)
</li>
<li>
phi4(r) = exp ( -0.5 r^2 / r0^2 ) (gaussian)
</li>
</ul>
Each uses a
"scale factor" r0, whose value is recommended to be greater than
the minimal distance between points, and rather less than the maximal distance.
Changing the value of r0 changes the shape of the interpolant function.
<p>
<p>
<b>RBF_INTERP_ND</b> needs access to the R8LIB library.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>RBF_INTERP_ND</b> is available in
<a href = "../../c_src/rbf_interp_nd/rbf_interp_nd.html">a C version</a> and
<a href = "../../cpp_src/rbf_interp_nd/rbf_interp_nd.html">a C++ version</a> and
<a href = "../../f77_src/rbf_interp_nd/rbf_interp_nd.html">a FORTRAN77 version</a> and
<a href = "../../f_src/rbf_interp_nd/rbf_interp_nd.html">a FORTRAN90 version</a> and
<a href = "../../m_src/rbf_interp_nd/rbf_interp_nd.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/lagrange_interp_nd/lagrange_interp_nd.html">
LAGRANGE_INTERP_ND</a>,
a C++ library which
defines and evaluates the Lagrange polynomial p(x)
which interpolates a set of data depending on a multidimensional argument x
that was evaluated on a product grid, so that p(x(i)) = z(i).
</p>
<p>
<a href = "../../cpp_src/r8lib/r8lib.html">
R8LIB</a>,
a C++ library which
contains many utility routines using double precision real (R8) arithmetic.
</p>
<p>
<a href = "../../cpp_src/rbf_interp_1d/rbf_interp_1d.html">
RBF_INTERP_1D</a>,
a C++ library which
defines and evaluates radial basis function (RBF) interpolants to 1D data.
</p>
<p>
<a href = "../../cpp_src/rbf_interp_2d/rbf_interp_2d.html">
RBF_INTERP_2D</a>,
a C++ library which
defines and evaluates radial basis function (RBF) interpolants to 2D data.
</p>
<p>
<a href = "../../cpp_src/shepard_interp_nd/shepard_interp_nd.html">
SHEPARD_INTERP_ND</a>,
a C++ library which
defines and evaluates Shepard interpolants to multidimensional data,
based on inverse distance weighting.
</p>
<p>
<a href = "../../cpp_src/sparse_interp_nd/sparse_interp_nd.html">
SPARSE_INTERP_ND</a>
a C++ library which
can be used to define a sparse interpolant to a function f(x) of a
multidimensional argument.
</p>
<p>
<a href = "../../m_src/spinterp/spinterp.html">
SPINTERP</a>,
a MATLAB library which
carries out piecewise multilinear hierarchical sparse grid interpolation;
an earlier version of this software is ACM TOMS Algorithm 847,
by Andreas Klimke;
</p>
<p>
<a href = "../../cpp_src/test_interp_nd/test_interp_nd.html">
TEST_INTERP_ND</a>,
a C++ library which
defines test problems for interpolation of data z(x),
depending on an M-dimensional argument.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Richard Franke,<br>
Scattered Data Interpolation: Tests of Some Methods,<br>
Mathematics of Computation,<br>
Volume 38, Number 157, January 1982, pages 181-200.
</li>
<li>
William Press, Brian Flannery, Saul Teukolsky, William Vetterling,<br>
Numerical Recipes in FORTRAN: The Art of Scientific Computing,<br>
Third Edition,<br>
Cambridge University Press, 2007,<br>
ISBN13: 978-0-521-88068-8,<br>
LC: QA297.N866.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "rbf_interp_nd.cpp">rbf_interp_nd.cpp</a>, the source code.
</li>
<li>
<a href = "rbf_interp_nd.sh">rbf_interp_nd.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "rbf_interp_nd_prb.cpp">rbf_interp_nd_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "rbf_interp_nd_prb.sh">rbf_interp_nd_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "rbf_interp_nd_prb_output.txt">rbf_interp_nd_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>DAXPY</b> computes constant times a vector plus a vector.
</li>
<li>
<b>DDOT</b> forms the dot product of two vectors.
</li>
<li>
<b>DNRM2</b> returns the euclidean norm of a vector.
</li>
<li>
<b>DROT</b> applies a plane rotation.
</li>
<li>
<b>DROTG</b> constructs a Givens plane rotation.
</li>
<li>
<b>DSCAL</b> scales a vector by a constant.
</li>
<li>
<b>DSVDC</b> computes the singular value decomposition of a real rectangular matrix.
</li>
<li>
<b>DSWAP</b> interchanges two vectors.
</li>
<li>
<b>PHI1</b> evaluates the multiquadric radial basis function.
</li>
<li>
<b>PHI2</b> evaluates the inverse multiquadric radial basis function.
</li>
<li>
<b>PHI3</b> evaluates the thin-plate spline radial basis function.
</li>
<li>
<b>PHI4</b> evaluates the gaussian radial basis function.
</li>
<li>
<b>R8MAT_SOLVE_SVD</b> solves a linear system A*x=b using the SVD.
</li>
<li>
<b>RBF_INTERP_ND</b> evaluates a radial basis function interpolant.
</li>
<li>
<b>RBF_WEIGHT</b> computes weights for radial basis function interpolation.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 05 October 2012.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>